-Everything is magic, just in different forms
(It is hinted that Thedas itself is a magical place, but it is so mundane to its residents that they don't realize that)
-Every mortal is a little bit magic, as it is what grants emotions and thoughts.
-However, what we typically define as a 'mage', is a mortal who has a greater connection to the realm of the Fade (in particular the Yellowish dream place of the Fade, as exploring elsewhere is either ancient history or considered impossible).
This measurement is called 'mana'. It is the spiritual part of a mage that may be depleted when casting magic.
-What mages typically utilize, is a sort of magic that taps into the greater realms beyond Thedas and brings it out into Thedas, using the mage as a conduit. A mage's staff is also used as a focus for this, even though it is not outright necessary - though with a staff focus, a mage can visualize a spell and refine it.
-There are other magics, of course. Since DAO there has been Blood Magic, which seems to operate in the form of self or external living sacrifice, and it seems more based on Thedas than on the Fade. In DAI there is the inclusion of Rift Magic (and a Spirit Magic tree that is related-looking to it in some ways), and this seems to draw more upon the magics of the realm of the Fade that may be called the 'Abyss', where the most bizarre magics are possible.
Essentially, mages are those who bring the Beyond into Thedas. They have that greater spark of the divine inside them, and are able to operate as bridges to and from the Fade. When they do enough, they can even weaken the Veil around them or the area that they've cast magic, and this opens them ever more to possession - at which point they'd need to dominate the demons they come across, if they want to continue casting such magic and being so open to the Fade. This binding of demons is something that Tevinter does, with Blood Magic, in order to keep them from overtaking their mages and society. This attaches demons to something of the physical (even a person) and has them attached to their will or non-will, keeping the demon's physical consistency but leaving it a slave.
To respond to earlier posts - all magic thins the Veil but in different ways. Blood Magic may reinforce a certain mode of existence for Thedas, but it breaches the Veil in violent ways. Other Magics simply weaken or warp the Veil. Rift Magic and related sorts may instead be what outright created the Veil (given that it isn't a literal Veil, but often can be considered a magic of 'perception', keeping Spirits and Mortals from interacting). Its important to remember that the Veil itself is magic, and that given the nature of Thedas so far, everything is magic and the greater illusion is actually that anything is 'real'.
Rift, Veil, Spirit, Creation magic may all have a sort of similar source, represented by a Green color.
Necromancy is a more newly introduced concept that may be considered akin to Blood Magic, but not exactly. Many spirits (minor spirits but typically demons) when crossing the Veil find corpses to inhabit, and we call these undead. Whereas normally it would (and it still can) require blood magic to bind these demons, necromancers instead utilize the specific magics of death (or the crossing between life and death) to still manipulate the dead and entropic energies, WITHOUT blood sacrifice/magic.
It is a still little-known practice that we may see elaborated when we get to Nevarra in a DLC/game. It is often represented by the Purple color, through I've seen Green and Red used for it too so a lot of stuff can be called 'necromancy' if its done through other means. But purple Necromancers do not sacrifice and do not create monstrosities, but instead do something that can be called more as a 'convincing' than a 'binding' of undead (compared to Blood Magic), getting them to do the same things as a full blood binding but without the literal attachment.
Magic is a mix of a lot of things. You may consider it more of a physical force, a way of nature. Or you may see it as drawing upon the Fade (or 'Beyond', as elves more consider it). Or you may see as ultimately a science in itself, that encompasses all things. Depends on your perspective and what you're using it for. In any case, it is what makes up the majority of Dragon Age's lore and plots, one way or another. Warriors still end up putting magical runes and gear on, or using magical derivatives themselves (like Templars and Reavers). Rogues still end up using magical concoctions and tap into a a part of themselves that just might be called magical (up for debate).
But so far, it is the Mages that directly interact with the magics of the world and thus they get a fair bit of the greater lore. Bioware balances this out by having Warriors more relevant for the 'epic battles' theme, and Rogues more relevant for the 'Thedas life' theme (whether its as a hunter in the wilds or a merc in the city), so we'll never get a Dragon Age that is 100% about the concept of magic.